http://ijsp.ccsenet.org International Journal of Statistics and Probability Vol. 7, No. 6; 2018 Copulas and Dependency Measures for Multivariate Linnik Distributions Alexandru Belu 1 , Wesley Maddox 2 & Wojbor A. Woyczynski 3 1 Bloomberg, LP., Financial Software Engineering, New York, NY, USA 2 Department of Statistical Science, Cornell University, Ithaca, NY 14853, USA 3 Department of Mathematics, Applied Mathematics and Statistics, and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44122, USA Correspondence: Wojbor A. Woyczynski, Department of Mathematics, Applied Mathematics and Statistics, and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44122, USA Received: September 24, 2018 Accepted: October 22, 2018 Online Published: October 31, 2018 doi:10.5539/ijsp.v7n6p154 URL: https://doi.org/10.5539/ijsp.v7n6p154 Abstract We begin this paper by introducing the Linnik distributions in both the univariate and multivariate case. An overview of simulation methods and two estimation procedures for the multivariate Linnik distribution are presented. Experiments demonstrating the accuracy of the procedures are also included. Then a novel multivariate Linnik copula is derived. The primary focus of this part of work is on simulation and estimation procedures for this copula, applying existing algorithms for simulation and estimation procedures for the multivariate Linnik distribution derived in the prior section. Several theoretical properties of the copula in relation to different dependence metrics are derived. Keywords: Copulas, α-stable processes, α- Linnik processes, estimation of copulas, dependence metrics 1. Introduction: Linnik Distributions In this section we recall the basic classical concepts related to L´ evy processes, and infinitely divisible distributions. Two special cases, one-dimensional and multivariate, are considered: the α-stable processes, and α-Linnik processes. Then the fundamental properties of the two classes are provided and compared with each other. In particular, the differences in the tail behavior of their distributions, and the fractional moment properties are explained. A study of the nonlinear evolution equations driven by the infinitesimal generators of Linnik processes can be found in Gunaratnam and Woyczynski(2015). 1.1 Preliminaries on General One-Dimensional L´ evy Processes and Infinitely Divisible Distributions A stochastic process X t , t ≥ 0, is called a L´ evy process (see, e.g, Bertoin (1996), Samorodnitsky and Taqqu (1994), and Sato (1999)) if : (i) X 0 = 0, with probability 1, (ii) X t has independent increments, i.e., for any n ∈ N, and any 0 < t 1 < t 2 < ··· < t n , the random variables X t 2 − X t 1 , X t 3 − X t 2 ,..., X t n − X t n−1 are independent, (iii) X t has stationary increments, i.e., for any t > s ≥ 0, X t − X s is equal in distribution to X t−s (iv) The trajectories of X t are right continuous with left limits, with probability 1. The 1-D distributions of the L´ evy processes are infinitely divisible (ID) which, in terms of the characteristics functions, can be expressed as the obvious equality: ϕ X (ξ, t) = E exp(iξ X(t)) = [ϕ X (ξ, t/n)] n ,ξ ∈ R. (1.1) for every n ≥ 1, t ≥ 0. A more detailed description of the structure of characteristic functions of infinitely divisible distributions is given by the following classical L´ evy-Khinchine Representation Theorem: THEOREM 2.1. A random variable X has an infinitely divisible distribution if, and only if, there exist µ ∈ R,σ ∈ R + , and a nonegative measure Λ on R\{0} satisfying the condition ∫ R (1 ∧| x| 2 )Λ(dx) < ∞, such that its characteristic function is of the form, ϕ X (ξ):= E[e iξX ] = e ψ(ξ) , (1.2) 154